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48 NATURE | VOL 399 | 6 MAY 1999 | www.nature.com after which it was cooled. Typically, during the dynamical evolution individual bonds undergo large-amplitude thermal ̄uctuations, which decrease on cooling. During the room temperature L ˆ 13:50 Ê A CPMD run for N ˆ 28, dissociation occurred at a bond location just outside the entrance to the knot, well separated from the terminal atom where the tension was applied. Figure 2 shows the region of the knot where the chain actually ruptured, both before (Fig. 2a) and after (Fig. 2b) the break. The change in hybridization of the C22 and C23 atoms from sp to sp is clearly evident also in Fig. 2c, where we display the time evolution of the system immediately before and after the break. Figure 3 shows a `snapshot' of the electronic charge density after the break occurred. The gap in the charge density between the C22 and C23 atoms con®rms the bond breaking at this position. Other simulations carried out with L > 13:50 Ê A yield breakpoints at one of the two bonds coming out from the knot. (Density-functional energy barriers are not quantitatively very accurate, but the general qualitative description of phenomena, as well as the dissociation energies involved in such processes, are known to be quite reliable.) The ®rst-principles CPMD calculations provide the total strain energy but not its pro®le along the chain. The latter can be obtained by making use of an ab initio force-constants model, which is able to reproduce the total strain energy to ,20%. Figure 4 shows the evolution of the strain energy distribution during structural relaxation of a N ˆ 30 trefoil just before breaking occurs. The strain energy is mainly localized in the two symmetric bonds that are outside the entrance to the knot. Our calculations suggest that 23 carbon atoms form the tightest knot that can be sustained in a polyethylene strand without it breaking. The strain energy stored in the knot at breaking point is 12.7 kcal mol per C±C bond, which is considerably smaller than the value (16.2 kcal mol) for the linear unknotted case. Thus, we ®nd that the presence of the knot has signi®cantly weakened the strand in which it is tied. Although the dynamical evolution of the present constrained ab initio MD simulations was suf®cient to observe bond breaking, no attempt was made to allow for recombination of the resulting radicals or reactions of the radicals with other parts of the chain. The study of these effects, as well as the role of chain branching and the in ̄uence of neighbouring chains, is left for future research. These factors will probably provide a deeper understanding of how the interplay between interand intra-molecular effects contributes to the mechanical properties of real polymer samples. Note added in proof : Arai et al. have recently reported the knotting of actin ®laments and DNA molecules using optical tweezers. They ®nd that the breaking stress for actin is signi®cantly lower than that of the unknotted ®laments, and that the breakage point is at the entrance to the knot, as our calculations predict. M